APEX Calculus

We are accustomed to describing surfaces as functions of two variables, usually written as $z=f(x,y)\text{.}$ For our coming needs, this method of describing surfaces will prove to be insufficient. Instead, we will parametrize our surfaces, describing them as the set of terminal points of some vector-valued function $\overrightarrow{r}(u,v)=⟨f(u,v),g(u,v),h(u,v)⟩\text{.}$ The bulk of this section is spent practicing the skill of describing a surface $\mathcal{S}$using a vector-valued function. Once this skill is developed, we’ll show how to find the surface area $S$ of a parametrically-defined surface $\mathcal{S}\text{,}$ a skill needed in the remaining sections of this chapter.

Given a point $(u_0,v_0)$ in the domain of a vector-valued function $\overrightarrow{r}\text{,}$ the vectors ${\overrightarrow{r}}_u(u_0,v_0)$ and ${\overrightarrow{r}}_v(u_0,v_0)$ are tangent to the surface $\mathcal{S}$ at $\overrightarrow{r}(u_0,v_0)$ (a proof of this is developed later in this section). The definition of smoothness dictates that ${\overrightarrow{r}}_u\times {\overrightarrow{r}}_v\ne \overrightarrow{0}\text{;}$ this ensures that neither ${\overrightarrow{r}}_u$ nor ${\overrightarrow{r}}_v$ are $\overrightarrow{0}\text{,}$ nor are they ever parallel. Therefore smoothness guarantees that ${\overrightarrow{r}}_u$ and ${\overrightarrow{r}}_v$ determine a plane that is tangent to $\mathcal{S}\text{.}$

A surface $\mathcal{S}$ is said to be orientable if a field of normal vectors can be defined on $\mathcal{S}$ that vary continuously along $\mathcal{S}\text{.}$ This definition may be hard to understand; it may help to know that orientable surfaces are often called “two sided.” A sphere is an orientable surface, and one can easily envision an “inside” and “outside” of the sphere. A paraboloid is orientable, where again one can generally envision “inside” and “outside” sides (or “top” and “bottom” sides) to this surface. Just about every surface that one can imagine is orientable, and we’ll assume all surfaces we deal with in this text are orientable.

It is enlightening to examine a classic non-orientable surface: the Möbius band, shown in Figure 15.5.3. Vectors normal to the surface are given, starting at the point indicated in the figure. These normal vectors “vary continuously” as they move along the surface. Letting each vector indicate the “top” side of the band, we can easily see near any vector which side is the “top”.

However, if as we progress along the band, we recognize that we are labeling “both sides” of the band as the top; in fact, there are not two “sides” to this band, but one. The Möbius band is a non-orientable surface.

We now practice parametrizing surfaces.

Examples 15.5.4 and 15.5.6 demonstrate an important principle when parametrizing surfaces given in the form $z=f(x,y)$ over a region $R\text{:}$ if one can determine $x$ and $y$ in terms of $u$ and $v\text{,}$ then $z$ follows directly as $z=f(x,y)\text{.}$

In the following two examples, we parametrize the same surface over triangular regions. Each will use $v$ as a “scaling factor” as done in Example 15.5.6.

When parametrizing the surface, we can let $x=u\text{,}$ $a\le u\le b\text{,}$ and we can let $y=g_1(u)+v(g_2(u)-g_1(u))\text{,}$ $0\le v\le 1\text{.}$ The parametrization of $x$ is straightforward, but look closely at how $y$ is determined. When $v=0\text{,}$ $y=g_1(u)=g_1(x)\text{.}$ When $v=1\text{,}$ $y=g_2(u)=g_2(x)\text{.}$

Following the above discussion, we can set $x=u\text{,}$ where $0\le u\le 3\text{,}$ and set $y=1+v(3-2u/3-1)=1+v(2-2u/3)\text{,}$ $0\le v\le 1\text{,}$ as used in that example.

One can do a similar thing if $R$ is bounded by $c\le y\le d\text{,}$ $h_1(y)\le x\le h_2(y)\text{,}$ but for the sake of simplicity we leave it to the reader to flesh out those details. The principles outlined above are given in the following Key Idea for reference.

Parametrization is a powerful way to represent surfaces. One of the advantages of the methods of parametrization described in this section is that the domain of $\overrightarrow{r}(u,v)$ is always a rectangle; that is, the bounds on $u$ and $v$ are constants. This will make some of our future computations easier to evaluate.

Just as we could parametrize curves in more than one way, there will always be multiple ways to parametrize a surface. Some ways will be more “natural” than others, but these other ways are not incorrect. Because technology is often readily available, it is often a good idea to check one’s work by graphing a parametrization of a surface to check if it indeed represents what it was intended to.

In Section 14.5 we used the area of a plane to approximate the surface area of a small portion of a surface. We will do the same here.

Let $R$ be the region of the $u$-$v$ plane bounded by $a\le u\le b\text{,}$ $c\le v\le d$ as shown in Figure 15.5.21.(a). Partition $R$ into rectangles of width $\mathrm{\Delta }u=\frac{b-a}{n}$ and height $\mathrm{\Delta }v=\frac{d-c}{n}\text{,}$ for some $n\text{.}$ Let $p=(u_0,v_0)$ be the lower left corner of some rectangle in the partition, and let $m$ and $q$ be neighboring corners as shown.

The point $p$ maps to a point $P=\overrightarrow{r}(u_0,v_0)$ on the surface $\mathcal{S}\text{,}$ and the rectangle with corners $p\text{,}$ $m$ and $q$ maps to some region (probably not rectangular) on the surface as shown in Figure 15.5.21.(b), where $M=\overrightarrow{r}(m)$ and $Q=\overrightarrow{r}(q)\text{.}$ We wish to approximate the surface area of this mapped region.

Let $\overrightarrow{u}=M-P$ and $\overrightarrow{v}=Q-P\text{.}$ These two vectors form a parallelogram, illustrated in Figure 15.5.21.(c), whose area approximates the surface area we seek. In this particular illustration, we can see that parallelogram does not particularly match well the region we wish to approximate, but that is acceptable; by increasing the number of partitions of $R\text{,}$ $\mathrm{\Delta }u$ and $\mathrm{\Delta }v$ shrink and our approximations will become better.

(This limit process also demonstrates that ${\overrightarrow{r}}_u(u,v)$ and ${\overrightarrow{r}}_v(u,v)$ are tangent to the surface $\mathcal{S}$ at $\overrightarrow{r}(u,v)\text{.}$ We don’t need this fact now, but it will be important in the next section.)

In Section 15.1, we recalled the arc length differential $ds=\left||\overrightarrow{r}{}^{\prime }(t)|\right|dt\text{.}$ In subsequent sections, we used that differential, but in most applications the “$\left||\overrightarrow{r}{}^{\prime }(t)|\right|$” part of the differential canceled out of the integrand (to our benefit, as integrating the square roots of functions is generally difficult). We will find a similar thing happens when we use the surface area differential $dS$ in the following sections. That is, our main goal is not to be able to compute surface area; rather, surface area is a tool to obtain other quantities that are more important and useful. In our applications, we will use $dS\text{,}$ but most of the time the “$\left||{\overrightarrow{r}}_u\times {\overrightarrow{r}}_v|\right|$” part will cancel out of the integrand, making the subsequent integration easier to compute.